For some systems that are not in equilibrium ie moving, D'Alembert introduced a method of applying a fictitious force to bring the system to equilibrium (rest). The equations of equilibrium could then be employed to solve the system. These equations then included the fictitious force.

Centrifugal force is just such a force applied to objects executing circular motion.

I think that there is no such thing as centrifugal force. ... is this force fictitious ?

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Depends on how you're using the term centrifucal force. There's the fictitious centrifugal force in a rotating frame. A less common (for physicists) usage is "reactive centrifugal force", which is part of a pair of Newton third law forces: say a string exerts a centripetal force on an object, then the object exerts a reactive outwards force on the string. Note that this "reactive centrifugal force" is not exerted on the object itself, but on the string. In a rotating frame, the fictitious forces, centripetal and coriolis (and euler if there is angular acceleration of the frame) all appear to act on the object. Wiki articles:

"Fictitious force" is just an unfortunate name choice. It gives people the wrong idea that some forces are more "real" than others. But that is just philosophy, irrelevant to physics. That is why I prefer the terms "inertial forces" and "interaction forces".

"Fictitious force" is just an unfortunate name choice. It gives people the wrong idea that some forces are more "real" than others. But that is just philosophy, irrelevant to physics. That is why I prefer the terms "inertial forces" and "interaction forces".

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Yes, I read this word "inertial" a lot on the Wikipedia link provided by rcgldr.

Before you get confusing explanations as to the different ways of regards motion under a central force or circular motion perhaps you would tell us what your course says ie which approach they take. Your exams will be based on this.

Before you get confusing explanations as to the different ways of regards motion under a central force or circular motion perhaps you would tell us what your course says ie which approach they take. Your exams will be based on this.

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Centripetal and centrifugal forces

I couldn't find the term centrifugal force in any of my physics books, hence I checked out PF. Can you provide some links (non Wikipedia) about centrifugal force .

Here is what I understand about this force :
I think this term was self created by humans for their own better understanding of the "motion" of objects in circular orbit.
We, the modern physicists would call it nothing but a consequence of the fact that any object undergoing circular motion tends to move in a straight line (tangential to its path) . So it can be thought of as "inertia" , but I would stick to my claim that this force is non existent.

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Yes, I read this word "inertial" a lot on the Wikipedia link provided by rcgldr. Can you explain this notion ?

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An "inertial" frame of reference means the frame of reference is not accelerating (no linear acceleration and no rotation). Newton's laws of physics apply in an inertial frame without any modification.

If the frame was accelerating, such as a rocket in space, then to an observer inside the rocket, there would be an apparent force on any object within the spacecraft, similar to gravity on earth.

So it can be thought of as "inertia" , but I would stick to my claim that this force is non existent.

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In your diagram of the car in a turn, there is a Newton third law pair of forces, an inwards centripetal force exerted by the road onto the car, and an equal in magnitude but opposing outwards reaction force that the car applies to the road. That outwards force is real, the issue for some people is using the term "reactive centrifugal force" to describe that outwards force that the car exerts to the road.

consider a train that is accelerating with acceleration a. You are on this train and you drop a ball. The only force due to a physical interaction acting on the ball is that of gravity which is due to the ball's interaction with the Earth.

However, you see the ball accelerating away from you with the same magnitude of acceleration as that of the acceleration of the train. Thus if we consider the acceleration of your reference frame (keep in mind that since you're measuring the motion of the ball with your position as the origin, you are at rest in your reference frame) as the source of a force on the ball, it applies a force of magnitude ma, where m is the mass of the ball. This force isn't due to any physical interaction that you can observe, that's why we call it fictitious. I agree that calling it fictitious is confusing and that inertial is less confusing and also an apt name in that the magnitude of the force is proportional to the mass of the object

A similar thing happens when you're in a rotating reference frame (for instance say someone does this while on a merry go round) except that now this fictitious force is directed outwards from the center of the merry go round. This type of fictious force due to being on an accelerated (or noninertial reference frame) is called centrifugal.

A centripetal force is the net force in the radial direction due to real forces and is generally talked about in an inertial or non-accelerated reference frame. Hope that clears it up a bit

Suppose you have a velocity vector [itex]\stackrel{\rightarrow}{U}[/itex] which is constant (no forces acting on it). That means

[itex] \dfrac{d}{dt}\stackrel{\rightarrow}{U} = 0 [/itex]

Now, this is a vector equation. It's common for people to denote vectors by ordered pairs:
[itex]\stackrel{\rightarrow}{U} = (U^x, U^y)[/itex], where [itex]U^x = \dfrac{dx}{dt}[/itex] and [itex]U^y = \dfrac{dy}{dt}[/itex].

But that's actually a BAD notation, because what it really means to write a vector as components is that we've chosen certain vectors called basis vectors, and we've expressed [itex]\stackrel{\rightarrow}{U}[/itex] as a combination of those basis vectors. So a better way to write it is

When you write it this way, you can see that
[itex] \dfrac{d}{dt}\stackrel{\rightarrow}{U} = \dfrac{dU^x}{dt} \hat{e_x} + U^x \dfrac{d \hat{e_x}}{dt} + \dfrac{dU^y}{dt} \hat{e_y} + U^y \dfrac{d \hat{e_y}}{dt}[/itex]

Of course, the most important fact about Cartesian coordinates is that the basis vectors are constant. So the terms [itex]\dfrac{d\hat{e_x}}{dt}[/itex] and [itex]\dfrac{d\hat{e_y}}{dt}[/itex] are both zero.

But if you switch to polar coordinates [itex]r,\theta[/itex], it's no longer the case that the basis vectors are constant. So you get additional terms having to do with the rate of change of the basis vectors. Those additional terms are the "fictitious forces".

I'm going to spare the details, but it turns out to be, in polar coordinates:

In your diagram of the car in a turn, there is a Newton third law pair of forces, an inwards centripetal force exerted by the road onto the car, and an equal in magnitude but opposing outwards reaction force that the car applies to the road. That outwards force is real, the issue for some people is using the term "reactive centrifugal force" to describe that outwards force that the car exerts to the road.

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If you think about it, the reactive force is just the force exerted by the car on to the road, but this does not relate (even remotely) to the diagram .
The diagram shows that there "seems" to be a force acting on the objects which tend to move outward (in the opposite direction of centripetal force). So the force you talked about is the force exerted by the car on the road and not the objects.

Think about this :

You are sitting on the backseat of a car, and this car is turning around a corner at high speed. You would have noticed yourself moving outwards (right ?) until you reach the door of the car which then would provide the appropriate centripetal force to your body.
Notice that this situation is based on the fact that friction could not provide the centripetal force. Notice also that at all times you would be "FEELING" some outward push , (again , which in my opinion is just inertia but not centrifugal force)

You switch frames back and forth and confuse the issue. Stick to an inertial frame, and there are no inertial forces. Stick to a non-inertial frame, and you always have inertial forces which act just like interaction forces (except Newtons 3rd).

Both is correct and leads to the same quantitative predictions, which is all that matters in physics. Musings about "realness of forces" is philosophy.

An "inertial" frame of reference means the frame of reference is not accelerating (no linear acceleration and no rotation). Newton's laws of physics apply in an inertial frame without any modification.

If the frame was accelerating, such as a rocket in space, then to an observer inside the rocket, there would be an apparent force on any object within the spacecraft, similar to gravity on earth.

In your diagram of the car in a turn, there is a Newton third law pair of forces, an inwards centripetal force exerted by the road onto the car, and an equal in magnitude but opposing outwards reaction force that the car applies to the road. That outwards force is real, the issue for some people is using the term "reactive centrifugal force" to describe that outwards force that the car exerts to the road.

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The centripetal force produces centripetal acceleration. The force that causes centripetal acceleration produces mechanical tensions (ie chains of electromagnetic forces at the atomic level). These mechanical tensions operate radially. The net sum of these tensions is the centripetal force that produces the centripetal acceleration.

Somebody had the idea that we should call the reaction to a centripetal force that is the result of the application mechanical forces a "centrifugal reaction force". As far as I can tell, this is a concept that serves no purpose and simply causes confusion. It can produce no acceleration so it is hardly a centre-fleeing force as the term "centrifugal" would imply. It is a mechanical tension so it does not exist where the centripetal force is applied by a force operating at a distance, such as gravity or an electric field.

Calling it a centripetal reaction force also confuses the physics. The true reaction to a centripetal force is another centripetal force. One does not appreciate this in the situation of a tethered ball rotating about a pole fixed to the earth but that is only because we do not take into account the movement of the earth. In the case of two balls tethered to each other by a rope and rotating in space about their centre of mass, the rope has a tension. Each ball pulls on the rope and, through the rope, pull on each other ie toward the centre. The ball pulls on the rope but produces no acceleration of the rope radially outward. Rather this pull is applied through the rope, via tensions, to accelerate the balls and rope toward the centre.

It can produce no acceleration so it is hardly a centre-fleeing force as the term "centrifugal" would imply.

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The term "centrifugal" has nothing to do with producing acceleration. It simply means that the force points away from the center of rotation. Acceleration is a matter of net force, not just one centrifugal force.